Deriving ols using M.o. M o The method of moments approach to estimation imp lio oposing the population moment restrictions on the sample moments o What does this mean? Recall that for E(X) the mean of a population distribution, a sample estimator of E(X) is simply the arithmetic mean of the sample Economics 20- Prof anderson
Economics 20 - Prof. Anderson 11 Deriving OLS using M.O.M. The method of moments approach to estimation implies imposing the population moment restrictions on the sample moments What does this mean? Recall that for E(X), the mean of a population distribution, a sample estimator of E(X) is simply the arithmetic mean of the sample
More derivation of ols We want to choose values of the parameters that will ensure that the sample versions of our moment restrictions are true o The sample versions are as follows n∑(,-A-Bx)=0 ∑x(,-B-Bx)=0 Economics 20- Prof anderson
Economics 20 - Prof. Anderson 12 More Derivation of OLS We want to choose values of the parameters that will ensure that the sample versions of our moment restrictions are true The sample versions are as follows: ( ) ( ˆ ˆ ) 0 ˆ ˆ 0 1 0 1 1 1 0 1 1 − − = − − = = − = − n i i i i n i i i n x y x n y x b b b b
More derivation of ols Given the definition of a sample mean, and properties of summation, we can rewrite the first condition as follows ,=Bo+B,x or B =y=Bx Economics 20- Prof anderson 13
Economics 20 - Prof. Anderson 13 More Derivation of OLS Given the definition of a sample mean, and properties of summation, we can rewrite the first condition as follows y x y x 0 1 0 1 ˆ ˆ or , ˆ ˆ b b b b = − = +
More derivation of ols ∑x(v-(-)Bx)=0 ∑x(-y)=∑x(x-x) ∑(x,-x)y-y)=B∑(x-x) Economics 20- Prof anderson 14
Economics 20 - Prof. Anderson 14 More Derivation of OLS ( ( ) ) ( ) ( ) ( )( ) ( ) = = = = = − − = − − = − − − − = n i i i n i i n i i i n i i i n i i i i x x y y x x x y y x x x x y y x x 1 2 1 1 1 1 1 1 1 1 ˆ ˆ 0 ˆ ˆ b b b b